Elizabeth and Olivia are saving their allowances. Elizabeth has 100 dollars in her account and saves 10 dollars per month. Olivia has 50 dollars in her account and saves 15 dollars per month. Write an equation that can be used to find 𝑚, the number of months until their accounts have the same balance.
Let’s firstly consider what we know about Elizabeth. She has 100 dollars in her account and saves 10 dollars per month. This means that an expression for the total amount of money she has in her account is 100 plus 10 multiplied by 𝑚, where 𝑚 is the number of months that she has been saving. Olivia, on the other hand, has 50 dollars in her account and saves 15 dollars per month. The expression for the amount of money in Olivia’s account after 𝑚 months is 50 plus 15 multiplied by 𝑚.
We want to write an equation where both girls have the same balance. Therefore, the two expressions need to be equal. 100 plus 10𝑚 has to be equal to 50 plus 15𝑚. This equation can be used to find 𝑚, the number of months until the accounts have the same balance. Whilst we have not been asked to solve the equation in this case, we can do so using a few steps.
Firstly, we can subtract 50 from both sides of the equation. 100 minus 50 is equal to 50. Therefore, the left-hand side simplifies to 50 plus 10𝑚. Subtracting 50 from the right-hand side leaves us with just 15𝑚. Our next step is to subtract 10𝑚 from both sides of this new equation. On the left-hand side, the 10𝑚s would cancel, leaving us with 50. 15𝑚 minus 10𝑚 is equal to five 𝑚. Therefore, the equation simplifies to 50 is equal to five 𝑚.
Our final step is to divide both sides of this equation by five. 50 divided by five is equal to 10. And, five 𝑚 divided by five is just equal to 𝑚. This means that the solution to the equation, 100 plus 10𝑚 is equal to 50 plus 15𝑚, is 𝑚 equals 10. We can therefore conclude that after 10 months, the girls will have the same balance. Elizabeth would have 100 dollars plus 10 multiplied by 10 dollars. This is equal to 200 dollars. Olivia would have 50 dollars plus 10 multiplied by 15 dollars. This is also equal to 200 dollars. Therefore, our answer is correct.